Application Of The Refined Arnoldi Method To The Calculation Of Critical Eigenvalues In Small Signal Stability Analysis

A power system is steady-state stable for a particular steady-state operating condition if, following any small disturbance, it reaches a steady-state operating condition which is identical or close to the pre-disturbance operating condition. This is also known as Small Signal Stability of a power system. In modern power system, Small Signal Stability is concerned with the oscillatory instability for lack of sufficient damping torque. Based on the State Space Modeling, Eigenvalue methods have shown to be a powerful tool in the analysis of electromechanical oscillatory. QR algorithm has been an effective way to calculate all the eigenvalues of space matrix. However, QR programs require a large amount of computer storage and CPU time for a large power system. To overcome the size limitation of the conventional eigenvalue programs, new techniques have been presented for calculating the eigenvalues associated with the electromechanical (rotor angle) modes. With the shift-invert transformation technology and implicit restarted theme, the refined Arnoldi method is applied to calculate critical eigenvalues in the study of small signal stability analysis in the thesis.The Arnoldi method is an orthogonal projection method onto a Krylov subspace. It starts with the Arnoldi procedure which builds an orthogonal basis of the Krylov subspace and reduces a large matrix into an upper Hessenberg one. A few Eigenvalues of the large matrix are obtained by iteratively calculating eigenvalues of the small one.The Ritz vectors obtained by Arnoldi method may not be good approximations and may not converge even if the corresponding Ritz values do. In order to overcome possible non-convergence of Ritz vectors, a refined strategy has been proposed. For each Ritz value, the refined method, instead of computing Ritz vectors, seek refined Ritz vector that minimizes the norm of residual formed with the Ritz values over the subspace involved. If the Ritz values converged, the refined Ritz vectors also converged to eigenvectors and they can be calculated with low cost.Eigen information of interest is the area near the imaginary axis with frequency ranging from 0.2 to 2 Hz. The modules of corresponding eigenvalues are small. However, Arnoldi algorithm is a dominant eigenvalue algorithm which is easy to calculate eigenvalues with the largest modules. So shift and invert transformation is used to transform critical eigenvalues to the largest ones with the eigenvectors remaining unchanged.Ritz values probably may not converge until the Krylov subspace dimension m gets very large. Extensive storage will be required and repeatedly finding the eigenvalues of small matrix also becomes expensive. The need to control the cost has motivated the development of implicit restarted scheme. It means replacing the starting vector with an "improved" starting vector and then computing a new Arnoldi factorization with the new vector.The implicit restarted technique is directly applied to the refined Arnoldi method. A useful shift strategy is presented by Jia and the best approximations of the unwanted Rtiz values are used as shifts (called refined shifts) at each restarting.The refined Arnoldi method is applied to the calculation of critical eigenvalues in small signal stability analysis in the thesis. State space matrix is gained by using the PMT modeling technique, and the eigenvalues near the shift point is obtained by using the implicitly restarted refined Arnoldi method (IRRA). Numerical experiment indicates that the refined Arnoldi method converge quickly and can calculate critical eigenvales of large power systems reliably and effectively.